Advance in Electronic and Electric Engineering. ISSN 31-197, Volume 3, Number (13), pp. 179-184 Research India Publications http://www.ripublication.com/aeee.htm Frequency Dependence Effective Refractive Index of Meta Materials by Effective Medium Theory G.N. Pandey 1, Arun Kumar 1 and Khem. B. Thapa 1 Department of Applied Physics, Amity Institute of Applied Sciences, Amity University, Noida (U.P.), Department of Physics, U I E T, Chhatrapati Shahu Ji Maharaj University, Kanpur- (UP), India. Abstract: In this present communication we studied the frequency dependence of the effective electromagnetic parameters of left-handed and related meta-materials (NIMs) of the split ring resonator (SRR) and wire type. By varying the width of inserted material as well as the host material, we also calculate the effective parameters of NIM like effective permittivity and effective permeability. Using these parameters, we calculate the effective refractive index and impedance of the considered medium. Our results reveal that the negative index of materials can be achieved when the thickness of the host materials should equal to the thickness of the inserted materials. OCIS codes: (35.438) Photonics; (16.3918) Metamaterial. 1. Introduction Electromagnetic meta-materials are artificially structured media with unique and distinct properties which are not observed in naturally occurring materials. More than three decades ago, Veselago [1] predicted many unusual properties of a hypothetical (at that time) isotropic medium with simultaneously negative electrical permittivity (ε) and magnetic permeability (μ) at frequency, which he named a left-handed material (LHM). As Veselago showed, LHMs display unique reversed electromagnetic (EM) properties as a result of an EM wave in such a medium having the triad k (wave vector), E (electric field), and H (magnetic field) left handed and, hence, exhibiting phase and energy velocities of opposite directions. It follows also that a LHM is
18 G.N. Pandey et al characterized by a negative refractive index (n); hence, their alternative name, negative-index materials (NIMs). These revolutionary properties open up a new regime in physics and technology (e.g. almost zero reflectivity at any angle of incidence), provided that such LHMs can be realized. Veselago s initial suggestion remained completely hypothetical, as naturally occurring materials do not provide such properties, until a significant breakthrough was announced in : Smith [] and coworkers presented evidence for a composite medium interlaced lattices of conducting rings and wires displaying negative values of ε and μ. For the development of this first LHM, Smith and his colleagues followed the pioneering work of Pendry et al. [3], who in 1999 developed designs for structures that are magnetically active, although made of non magnetic materials. One of those structures is the so-called split-ring resonator (SRR) structure, composed of metallic rings with gaps which have been widely adopted as the model for creating negative μ at gigahertz frequencies.. Theory and Physical Model To study the permittivity and permeability in such medium we have different theories for continuous as well as periodic structures. There are continuous as well as periodic (artificial) materials as shown in Fig. A series of polarizable planar sheets, of width l, is spaced apart with periodicity d. We assume that an electromagnetic wave propagates in the direction along the normal to the sheets, with fields polarized in the plane of the sheets. The wave propagation behavior under these assumptions reduces to a onedimensional boundary-value problem, in which solutions to the wave equation in three regions must be considered: -d/ z<-l/,-l/ z<l/ and l/ z<d/. The dispersion relation and fields for the one-dimensional periodic system can be determined through the use of the transfer-matrix technique. The one dimensional (1D) transfer matrix relates the fields on one side of a planar slab to the other. Fig. 1: Periodic system of electrically or magnetically polarizable sheets. The sheets have thickness l and are spaced a distance d apart. An electromagnetic wave is assumed to propagate along the direction normal to the sheets.
Frequency Dependence Effective Refractive Index of Meta Materials 181 The transfer matrix can be defined from F ' TF, where F E (1) E and H red are the complex electric and magnetic field amplitudes located on the right-hand (unprimed) and lefthand(primed) faces of the slab. The magnetic field here is a reduced magnetic field having the normalization H red +iωμ H. The transfer matrix for a homogeneous 1D slab has the analytic form, z cos( nkd) sin( nkd) T k () k sin( nkd) cos( nkd) z where n is the refractive index and z is the wave impedance of the slab. Note that n and z are related to the local relative permittivity and permeability in the usual manner n εμ, z μ / ε (3) The fields on the two sides of a unit cell that is composed of three distinct planar regions vacuum/material/vacuum can be related by a transfer matrix that consists of the matrix product of the transfer matrices in each region or T T T T (4) tot vacuum material vacuum H red where T vacuum kd cos kd k sin 1 kd sin k kd cos z cos( nkl) sin( nkl) and T k material (5) k sin( nkl) cos( nkl) z Using the transfer-matrix formalism with periodic boundary conditions, the following expressions for the propagation factor and Bloch impedance is obtained: cos( α d ) T + T and 11 T1 Z B, red (6) T1 where the latter expression is true provided the unit cell possesses reflection symmetry in the direction of propagation. The constitutive parameters determined by using eq.(6) with eq.(3) are termed the Bloch constitutive parameters [4].
18 G.N. Pandey et al 3. Results and Discussion The Maxwell s equations show that the refractive index is dependent upon the electric permittivity and magnetic permeability of the materials. In 1968 Veselago[1] already mentioned on the basis of calculation that the negative refractive index is possible if the electric permittivity and magnetic permeability are both simultaneously negative at certain frequency [1]. The period structure or photonic crystals are invented that such photonic crystals have the negative refractive index due to anomalous dispersion relation. This prediction has opened all the possibility for the existence of the negative refractive index and they mentioned the artificial structure possible only with negative permittivity and negative permeability [5]. The researcher have made the artificial structure of SRR at microwave frequency which has negative refractive index due to the following parameters of the SRR [6], like ε ( ω) H ωep ωe 1 ω ωe + iγω, μ ( ω) H ωmp ωe 1 ω ωm + iγω. In Fig., we have plotted electric permittivity (ε), magnetic permeability (μ) and refractive index of the SRR materials versus angular frequency (ω). Permittivity 5 x 1-3 -5 4 6 8 1 1 14 16 18 Permeability 5 x 1-3 -5 4 6 8 1 1 14 16 18 Refractive Index x 1-3 - 4 6 8 1 1 14 16 18 Angular Frequency (ω) Fig. : Electric permittivity, magnetic permeability and refractive index of SRR versus angular frequency.
Frequency Dependence Effective Refractive Index of Meta Materials 183 Finally, we set the width of both the host material and inserted material (L and d) same, i. e. L1mm and d1mm, we obtained surprisingly interesting result for the range of frequency from Hz to Hz. These results are shown in figure 3. The first part of Fig. 3 shows that electric permittivity (ε) remains positive before 11 Hz, at 11 Hz it becomes zero and afterwards it starts tending towards negative value, whereas the second part of fig.3 shows upto 165 Hz, the magnetic permeability (µ) remains positive at 165 Hz it becomes zero and afterwards it start tending towards more and more negative value. So in accordance with equation 3, the n eff before 11 Hz remains positive, at 11 Hz it becomes zero. In the range from 11 Hz to 165 Hz the real part of magnetic permeability (µ) become zero and the complex nature of refractive index(n) appears only imaginary part of n is obtained in this range. The fourth part of this figure shows how impedance responds to this frequency range. Since at 11 Hz electric permittivity(ε) is zero so there is a peak at 11 Hz, after 11 Hz the impedance becomes imaginary upto 165 Hz. Beyond 165 Hz impedance (Z eff ) becomes zero. We conclude the effective permeability and permittivity of the NIM materials is affected with thickness of the medium and such study is used to fabricate the NIM artificial materials. x 1-3 4 x 1-3 Permittivity - Permeability -4 5 1 15-5 1 15 Effective Index x 1-3 1-1 - 5 1 15 Angular Frequency (ω) Effective Impedance 5 1 15 Angular Frequency (ω) Fig. 3: Electric permittivity, magnetic permeability and refractive index of composite material versus angular frequency on taking L1mm; d1mm. 4 3 1 References [1] V. G. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ, Sov Phys., Usp 1:59 514, 1968.
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